Figure 1.
Optimal control $ u_{1}(t) $ .
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July
2020, 16(4): 1663-1683.
doi: 10.3934/jimo.2019023
Minimizing almost smooth control variation in nonlinear optimal control problems
| 1. | Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China |
| 2. | Department of Mathematics, Shanghai University, Baoshan 200444, Shanghai, China |
| 3. | Xingzhi College, Zhejiang Normal University, Jinhua 321004, Zhejiang, China |
In this paper, we consider an optimal control problem in which the control is almost smooth and the state and control are subject to terminal state constraints and continuous state and control inequality constraints. By introducing an extra set of differential equations for this almost smooth control, we transform this constrained optimal control problem into an equivalent problem involving both control function and system parameter vector as decision variables. Then, by the control parametrization technique and a time scaling transformation, the equivalent problem is approximated by a sequence of constrained optimal parameter selection problems, each of which is a finite dimensional optimization problem. For each of these constrained optimal parameter selection problems, a novel exact penalty function method is constructed by appending penalized constraint violations to the cost function. This gives rise to a sequence of unconstrained optimal parameter selection problems; and each of which can be solved by existing optimization algorithms or software packages. Finally, a practical container crane operation problem is solved, showing the effectiveness and applicability of the proposed approach.
Mathematics Subject Classification:
Primary: 58F15, 58F17; Secondary: 53C35.
Citation:
Ying Zhang, Changjun Yu, Yingtao Xu, Yanqin Bai. Minimizing almost smooth control variation in nonlinear optimal control problems. Journal of Industrial and Management Optimization,
2020, 16
(4)
: 1663-1683.
doi: 10.3934/jimo.2019023
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Inexact restoration and adaptive mesh refinement for optimal control, Journal of Industrial and Management Optimization, 10 (2014), 521-542.
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D. Chang and Z. Wu,
Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance, Journal of Industrial and Management Optimization, 11 (2015), 27-40.
doi: 10.3934/jimo.2015.11.27. |
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M. Gerdts and M. Kunkel,
A nonsmooth Newton's method for discretized optimal control problems with state and control constraints, Journal of Industrial and Management Optimization, 4 (2008), 247-270.
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Determining the viability for hybrid control systems on a region with piecewise smooth boundary, Numerical Algebra, Control and Optimization, 5 (2015), 1-9.
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B. Li, C. Xu, K. L. Teo and J. Chu,
Time optimal Zermelo's navigation problem with moving and fixed obstacles, Applied Mathematics and Computation, 224 (2013), 866-875.
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Q. Lin, R. Loxton and K. L. Teo,
The control parameterization method for nonlinear optimal control: A surney, Journal of Industrial and Management Optimization, 10 (2014), 275-309.
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R. Loxton, Q. Lin, V. Rehbock and K. L. Teo,
Control parameterization for optimal control problems with continuous inequality constraints: New convergence results, Numerical Algebra, Control and Optimization, 2 (2012), 571-599.
doi: 10.3934/naco.2012.2.571. |
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K. S. Peterson and A. G. Stefanopoulou,
Extremum seeking control for soft landing of an electromechanical valve actuator, Automatica, 40 (2004), 1063-1069.
doi: 10.1016/j.automatica.2004.01.027. |
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V. Rehbock, Tracking Control and Optimal Control, PhD thesis, University of Western Australia, Perth, 1994. |
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Y. Sakawa and Y. Shindo,
Optimal control of container cranes, Automatica, 18 (1982), 257-266.
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K. Schittkowski,
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K. L. Teo and L. S. Jennings,
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doi: 10.1007/BF00940727. |
| [20] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Long-man Scientific and Technical, Essex, 1991. |
| [21] |
W. X. Wang, Y. L. Shang, L. S. Zhang, et. al., Global minimization of non-smooth unconstrained problems with filled function, Optimization Letters, 7 (2013), 435-446.
doi: 10.1007/s11590-011-0427-7. |
| [22] |
W. Xu, Z. G. Feng, J. W. Peng and K. F. C. Yiu,
Optimal switching for linear quadratic problem of switched systems in discrete time, Automatica, 78 (2017), 185-193.
doi: 10.1016/j.automatica.2016.12.002. |
| [23] |
K. F. C. Yiu, Y. Liu and K. L. Teo,
A hybrid descent method for global optimization, Journal of Global Optimization, 28 (2004), 229-238.
doi: 10.1023/B:JOGO.0000015313.93974.b0. |
| [24] |
C. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai,
A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial and Management Optimization, 6 (2010), 895-910.
doi: 10.3934/jimo.2010.6.895. |
| [25] |
C. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai,
On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, Journal of Industrial Management and Optimization, 8 (2012), 485-491.
doi: 10.3934/jimo.2012.8.485. |
show all references
References:
| [1] |
N. U. Ahmed, Dynamic Systems and Control with Applications, Singapore: World Scientific, 2006.
doi: 10.1142/6262. |
| [2] |
N. Banihashemi and C. Y. Kaya,
Inexact restoration and adaptive mesh refinement for optimal control, Journal of Industrial and Management Optimization, 10 (2014), 521-542.
doi: 10.3934/jimo.2014.10.521. |
| [3] |
D. Chang and Z. Wu,
Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance, Journal of Industrial and Management Optimization, 11 (2015), 27-40.
doi: 10.3934/jimo.2015.11.27. |
| [4] |
M. Gerdts and M. Kunkel,
A nonsmooth Newton's method for discretized optimal control problems with state and control constraints, Journal of Industrial and Management Optimization, 4 (2008), 247-270.
doi: 10.3934/jimo.2008.4.247. |
| [5] |
L. G |
| [6] |
Y. Han and Y. Gao,
Determining the viability for hybrid control systems on a region with piecewise smooth boundary, Numerical Algebra, Control and Optimization, 5 (2015), 1-9.
doi: 10.3934/naco.2015.5.1. |
| [7] |
L. S. Jennings, M. E. Fisher, K. L. Teo, et al., MISER 3 Optimal Control Software: Theory and User Manual, version 3, University of Western Australia, 2004. |
| [8] |
L. Jennings, C. Yu, B. Li, V. Rehbock, R. Loxton and F. Yang,
Visual miser: an efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2015), 781-810.
doi: 10.3934/jimo.2016.12.781. |
| [9] |
J. Kaartinen, J. H |
| [10] |
C. T. Lawrence and A. L. Tits,
A computationally efficient feasible sequential quadratic programming algorithm, SIAM Journal on Optimization, 11 (2006), 1092-1118.
doi: 10.1137/S1052623498344562. |
| [11] |
H. W. J. Lee, K. L. Teo, V. Rehbock and L. S. Jennings,
Control parametrization enhancing technique for time optimal control problems, Dynamic Systems and Applications, 6 (1997), 243-261.
|
| [12] |
B. Li, C. Xu, K. L. Teo and J. Chu,
Time optimal Zermelo's navigation problem with moving and fixed obstacles, Applied Mathematics and Computation, 224 (2013), 866-875.
doi: 10.1016/j.amc.2013.08.092. |
| [13] |
Q. Lin, R. Loxton and K. L. Teo,
The control parameterization method for nonlinear optimal control: A surney, Journal of Industrial and Management Optimization, 10 (2014), 275-309.
doi: 10.3934/jimo.2014.10.275. |
| [14] |
R. Loxton, Q. Lin, V. Rehbock and K. L. Teo,
Control parameterization for optimal control problems with continuous inequality constraints: New convergence results, Numerical Algebra, Control and Optimization, 2 (2012), 571-599.
doi: 10.3934/naco.2012.2.571. |
| [15] |
K. S. Peterson and A. G. Stefanopoulou,
Extremum seeking control for soft landing of an electromechanical valve actuator, Automatica, 40 (2004), 1063-1069.
doi: 10.1016/j.automatica.2004.01.027. |
| [16] |
V. Rehbock, Tracking Control and Optimal Control, PhD thesis, University of Western Australia, Perth, 1994. |
| [17] |
Y. Sakawa and Y. Shindo,
Optimal control of container cranes, Automatica, 18 (1982), 257-266.
|
| [18] |
K. Schittkowski,
NLPQL: A fortran subroutine solving constrained nonlinear programming problems, Ann. Oper. Res., 5 (1986), 485-500.
doi: 10.1007/BF02739235. |
| [19] |
K. L. Teo and L. S. Jennings,
Nonlinear optimal control problems with continuous state inequality constraints, Journal of Optimization Theory and Application, 63 (1989), 1-22.
doi: 10.1007/BF00940727. |
| [20] |
K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Long-man Scientific and Technical, Essex, 1991. |
| [21] |
W. X. Wang, Y. L. Shang, L. S. Zhang, et. al., Global minimization of non-smooth unconstrained problems with filled function, Optimization Letters, 7 (2013), 435-446.
doi: 10.1007/s11590-011-0427-7. |
| [22] |
W. Xu, Z. G. Feng, J. W. Peng and K. F. C. Yiu,
Optimal switching for linear quadratic problem of switched systems in discrete time, Automatica, 78 (2017), 185-193.
doi: 10.1016/j.automatica.2016.12.002. |
| [23] |
K. F. C. Yiu, Y. Liu and K. L. Teo,
A hybrid descent method for global optimization, Journal of Global Optimization, 28 (2004), 229-238.
doi: 10.1023/B:JOGO.0000015313.93974.b0. |
| [24] |
C. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai,
A new exact penalty function method for continuous inequality constrained optimization problems, Journal of Industrial and Management Optimization, 6 (2010), 895-910.
doi: 10.3934/jimo.2010.6.895. |
| [25] |
C. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai,
On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, Journal of Industrial Management and Optimization, 8 (2012), 485-491.
doi: 10.3934/jimo.2012.8.485. |
Figure 2.
Optimal control function $ u_{2}(t) $ .
Figure 3.
Optimal state trajectory $ x_{1}(t) $ .
Figure 4.
Optimal state trajectory $ x_{2}(t) $ .
Figure 5.
Optimal control $ x_{3}(t) $ .
Figure 6.
Optimal state trajectory $ x_{4}(t) $ .
Figure 7.
Optimal state trajectory $ x_{5}(t) $ .
Figure 8.
Optimal control $ x_{6}(t) $ .
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